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%%文档的题目、作者与日期
\author{王立庆（2019级数学与应用数学1班）}
\title{数量金融实验 - 第4章课堂练习}
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%\date{2022 年 9 月 8 日}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item %1
期权定价的最基本的 Black-Scholes 模型的假设中，下述那条不在其中？
\begin{enumerate}
\item[A.]  股票价格服从几何布朗运动。
\item[B.]  无交易费用和税收。
\item[C.]  无风险利率是常数，股票不支付股息。
\item[D.]  投资者不能按无风险利率任意地借入或贷出。
\end{enumerate}

{\color{red}
解答：D. 最基本的BS 模型中，投资者可以按无风险利率任意地借入或贷出。

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\item %2
在 BS 模型中，无风险资产、风险资产和期权的演化过程满足的方程中，不正确的是哪个？
\begin{enumerate}
\item[A.]  股票价格满足方程 $dS_t = \mu S_t dt + \sigma S_t dW_t$. 
\item[B.]  债券价格满足方程 $dB_t = rdt$.
\item[C.]  期权价格满足方程 $dV_t = dV(S_t,t)$. 
\item[D.]  期权价格满足方程 $dV_t = \left( \frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} 
+ \mu S \frac{\partial V}{\partial S} \right)dt + \sigma S \frac{\partial V}{\partial S} dW_t $. 
\end{enumerate}

{\color{red}
解答：B. 债券价格满足方程 $dB_t = rB_tdt$.

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\item %3
使用股票与期权的一个投资组合来导出 BS 方程，不正确的说法是哪个？
\begin{enumerate}
\item[A.]  期权价格是股票价格和时间的函数。
\item[B.]  期权价格的增量包含确定部分和随机部分。
\item[C.]  投资组合的收益增量由每份股票和每份债券的价格增量得出。
\item[D.]  根据无套利原理，投资组合的收益增量等于无风险资产的收益增量，从而得出 BS 方程。
\end{enumerate}

{\color{red}
解答：C. 投资组合的收益增量由每份股票和每份期权的价格增量得出。

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\item %4
下述哪个是正确的 BS 方程？
\begin{enumerate}
\item[A.]  $\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV =0$.
\item[B.]  $\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 V}{\partial S^2} + r \frac{\partial V}{\partial S} - rV =0$.
\item[C.]  $t\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV =0$.
\item[D.]  $t\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 V}{\partial S^2} + r \frac{\partial V}{\partial S} - rV =0$.
\end{enumerate}

{\color{red}
解答：A. 期权价格关于股票价格的偏导数前面，都有股票价格作为乘法因子。

}

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\item %5
设 $V=V(S,t)$ 是二元函数，设自变量变换 $x=\ln S$, 则 $\frac{\partial V}{\partial S}$ 与 $\frac{\partial V}{\partial x}$ 的关系是什么？
\begin{enumerate}
\item[A.]  $ S\frac{\partial V}{\partial S} = \frac{\partial V}{\partial x}$. 
\item[B.]  $ x\frac{\partial V}{\partial S} = \frac{\partial V}{\partial x}$. 
\item[C.]  $ \frac{\partial V}{\partial S} = S\frac{\partial V}{\partial x}$. 
\item[D.]  $ \frac{\partial V}{\partial S} = x\frac{\partial V}{\partial x}$. 
\end{enumerate}

{\color{red}
解答：A. 根据链式法则即得。

}

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\item %6
设 $v=v(x,t)$ 是二元函数，设应变量变换 $v(x,t) = u(x,t)e^{at+bx}$, 则 $\frac{\partial v}{\partial x}$ 与 $\frac{\partial u}{\partial x}$ 的关系是什么？
\begin{enumerate}
\item[A.]  $\frac{\partial v}{\partial x} - bv = \frac{\partial u}{\partial x} e^{at+bx}$. 
\item[B.]  $\frac{\partial v}{\partial x} + bv = \frac{\partial u}{\partial x} e^{at+bx}$. 
\item[C.]  $\frac{\partial v}{\partial x} - bv = \frac{\partial u}{\partial x}$. 
\item[D.]  $\frac{\partial v}{\partial x} + bv = \frac{\partial u}{\partial x}$. 
\end{enumerate}

{\color{red}
解答：A. 根据链式法则即得。

}

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\item %7
热传导方程 $\frac{\partial u}{\partial t} = a^2\frac{\partial^2 u }{\partial x^2}, u(x,0)=\Phi(x)$ 的通解是什么？
\begin{enumerate}
\item[A.]  $u(x,t) = \frac{1}{2a\sqrt{\pi t}} \int_{-\infty}^{\infty} \exp\left( - \frac{(x-\xi)^2}{4a^2}\right) \Phi(\xi)d\xi$. 
\item[B.]  $u(x,t) = \frac{1}{2a\sqrt{\pi}t} \int_{-\infty}^{\infty} \exp\left( - \frac{(x-\xi)^2}{4a^2}\right) \Phi(\xi)d\xi$.
\item[C.]  $u(x,t) = \frac{1}{2a\sqrt{\pi t}} \int_{-\infty}^{\infty} \exp\left( - \frac{(x-\xi)^2}{4a^2t}\right) \Phi(\xi)d\xi$.
\item[D.]  $u(x,t) = \frac{1}{2a\sqrt{\pi}t} \int_{-\infty}^{\infty} \exp\left( - \frac{(x-\xi)^2}{4a^2t}\right) \Phi(\xi)d\xi$.
\end{enumerate}

{\color{red}
解答：C. 根据泊松公式可得。

}

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\item %8
有关 Radon-Nikodym 导数的叙述，下述说法中，不正确的是哪个？
\begin{enumerate}
\item[A.]  可测空间是指二元组 $(\Omega, \mathcal{F})$, 其中 $\Omega$ 是一个集合，$\mathcal{F}$ 是 $\Omega$ 中的一些子集组成的$\sigma$-域。
\item[B.]  称 $P$ 是可测空间 $(\Omega, \mathcal{F})$ 上的一个测度，是指对每个 $A\in\mathcal{F}$, 它的测度 $P(A)$ 都有定义，且满足可列可加等条件。
\item[C.]  RN 导数比较了同一个可测空间上的两个测度 $P$ 与 $Q$ 的商。
\item[D.]  RN 导数是集合 $\Omega$ 上的一个非负函数 $f(\omega)$, 使得对任意子集 $A\subseteq \Omega$, 都有 $Q(A) = \int_A f(\omega) dP(\omega)$. 
\end{enumerate}

{\color{red}
解答：D. RN 导数是集合 $\Omega$ 上的一个非负函数 $f(\omega)$, 使得对任意 $A\in \mathcal{F}$, 都有 $Q(A) = \int_A f(\omega) dP(\omega)$. 

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\item %9
设 $(\Omega,\mathcal{F},P)$ 是一个概率空间，设 $X$ 是一个 $P$-可测的随机变量，在 $P$ 测度下服从标准正态分布。通过 RN 导数 $$\frac{dQ}{dP} = \exp\left( -\frac{\theta^2}{2} - \theta x \right)$$
定义另一个测度 $Q$. 则随机变量 $Y=X+\theta$ 在测度 $Q$ 下服从什么分布？

\begin{enumerate}
\item[A.]  $N(\theta, 1)$. 
\item[B.]  $N(0,1)$.
\item[C.]  $N(-\theta,1)$. 
\item[D.]  $N(0,\theta^2)$. 
\end{enumerate}

{\color{red}
解答：B. 计算可得 $Q(Y\le y)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^y \exp\left( -\frac{z^2}{2} \right) dz$.  
所以随机变量 $Y$ 在测度 $Q$ 下服从标准正态分布。 

}

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\item %9
关于风险管理中的希腊字母，下述说法中，不正确的是哪个？
\begin{enumerate}
\item[A.]  $\Delta$ 是期权价格对标的资产价格的变化率。
\item[B.]  $\Gamma$ 是期权价格对标的资产价格的二阶导数。
\item[C.]  $\Theta$ 是期权价格对时间的变化率。
\item[D.]  $\rho$ 是期权价格对标的资产的波动率的变化率。
\end{enumerate}

{\color{red}
解答：D. $\rho$ 是期权价格对无风险利率的变化率。
 

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\end{enumerate}

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